Grand canonical Partition function of fluids

Question

I'm going through the book Lectures on Phase transitions and Renormalization Group by Nigel Goldenfeld. In chapter two section 2.11 (Fluids), the grand partition function describing the fluid is :

$\mathcal{\Theta} $= Tr $e^{-\beta[\mathcal{H} - \mu N]} $ , with Tr = $\sum_{N=0}^{\infty} \frac{1}{N!}\int \prod_i ^N \frac{d^dp_id^dr_i}{h^{dN}}$.

My initial question is regarding the sum over N. What I assumed is that since we are in a grand canonical ensemble, he is doing the sum over all energy configurations of the system, therefore we sum over all possible momentum, position and N(due to the fact that we might have configurations with different number of particles considered "inside" our system). However, even if what I said is the true, I still don't understand why the N! term appears (nor the $h^{dN}$).

Going through my old lecture notes of stat mech and through the reference books I wasn't able to find a rigorous way in which this form of the partition function appears. Thus I ask you to help me better grasp this concept. I would also be pleased if you could share the name of some books that go through this procedure in detail.

Another question that I have is about the use of the trace. In the 1-d Ising model, due to the fact that we use transfer matrices we were able to replace the $\sum_{configuraions}$ by Trace. However, I believed it was the case only for the 1d ising model. Under which argument did he use the fact that Trace could replace the sum ? (What I assumed is that he just wrote Tr out of habit, because he later defines it as being a sum and interals and it is not the Trace in its actual sense)

Answer 1

I'd suggest reading chapter 4 of McGreevy's notes (MIT prof): https://mcgreevy.physics.ucsd.edu/s12/lecture-notes/chapter04.pdf

This partition function, which includes the terms $N!$ and $h^{dn}$ is used in the derivation of the Sackur-Tetrode equation. Although this equation is much more fundamental than the phase transitions you are reading about, the idea is the same: when counting states, we over count by a factor of $N!$ when not accounting for the indistinguishable nature of those particles.

The $h^{dn}$ factor comes about because we need the state counting function to be dimensionless. Some writers allude to the idea that the Planck constant enters due to its role in defining the smallest regions of space. Experimentation does show that this factor ($h$) is correct.